= Monatomic gas
{title2=$\gamma=5/3$}
A monatomic gas consists of individual atoms or ions rather than molecules. In the classical dilute ideal-gas regime, its three translational degrees of freedom give $U=3Nk_BT/2$ by <equipartition theorem>. Thus $C_V=3Nk_B/2$, while $C_P-C_V=Nk_B$ follows by differentiating the ideal-gas enthalpy $H=U+PV=5Nk_BT/2$. Consequently the <adiabatic index> is $\gamma=C_P/C_V=5/3$ and the <specific internal energy> is $e=3P/(2\rho)$. The same translational result applies to a fully ionized mixture when the particle numbers are fixed. This excludes additional excitation, ionization and radiation contributions to the heat capacity.
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